?gglse

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?gglse

Solves the linear equality-constrained least squares problem using a generalized RQ factorization.

Syntax

call sgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)

call dgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)

call cgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)

call zgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)

call gglse(a, b, c, d, x [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine solves the linear equality-constrained least squares (LSE) problem:

minimize ||c - A*x||² subject to B*x = d

where A is an m-by-n matrix, B is a p-by-n matrix, c is a given m-vector, andd is a given p-vector. It is assumed that p≤n≤m+p, and

These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by

B=(0 R)*Q, A=Z*T*Q

Input Parameters

m

INTEGER. The number of rows of the matrix A (m≥ 0).

n

INTEGER. The number of columns of the matrices A and B (n≥ 0).

p

INTEGER. The number of rows of the matrix B

(0 ≤p≤n≤m+p).

a, b, c, d, work

REAL for sgglse

DOUBLE PRECISION for dgglse

COMPLEX for cgglse

DOUBLE COMPLEX for zgglse.

Arrays:

a(lda,*) contains the m-by-n matrix A.

The second dimension of a must be at least max(1, n).

b(ldb,*) contains the p-by-nmatrix B.

The second dimension of b must be at least max(1, n).

c(*), size at least max(1, m), contains the right hand side vector for the least squares part of the LSE problem.

d(*),, size at least max(1, p), contains the right hand side vector for the constrained equation.

work is a workspace array, its dimension max(1, lwork).

lda

INTEGER. The leading dimension of a; at least max(1, m).

ldb

INTEGER. The leading dimension of b; at least max(1, p).

lwork

INTEGER. The size of the work array;

lwork≥ max(1, m+n+p).

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

See Application Notes for the suggested value of lwork.

Output Parameters

a

The elements on and above the diagonal contain the min(m, n)-by-n upper trapezoidal matrix T as returned by ?ggrqf.

x

REAL for sgglse

The solution of the LSE problem.

b

On exit, the upper right triangle of the subarray b(1:p, n-p+1:n) contains the p-by-p upper triangular matrix R as returned by ?ggrqf.

d

On exit, d is destroyed.

c

On exit, the residual sum-of-squares for the solution is given by the sum of squares of elements n-p+1 to m of vector c.

work(1)

If info = 0, on exit, work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = 1, the upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is singular, so that rank(B) < p; the least squares solution could not be computed.

If info = 2, the (n-p)-by-(n-p) part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is singular, so that

; the least squares solution could not be computed.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gglse interface are the following:

a: Holds the matrix A of size (m,n).
b: Holds the matrix B of size (p,n).
c: Holds the vector of length (m).
d: Holds the vector of length (p).
x: Holds the vector of length n.

Application Notes

For optimum performance, use

lwork≥p + min(m, n) + max(m, n)*nb,

where nb is an upper bound for the optimal blocksizes for ?geqrf, ?gerqf, ?ormqr/?unmqr and ?ormrq/?unmrq.

You may set lwork to -1. The routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Generalized Linear Least Squares (LLS) Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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